Vector Geometry

Review question

The position vectors of four points $A, B, C, D$ relative to an origin $O$ are $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ respectively. Obtain the equation of the line $l$ joining the midpoints of $AB$ and $CD$. Show that $l$ intersects the line joining the midpoints of $AC$ and $BD$ and find the position vector $\mathbf{p}$ of the point $P$ of intersection.
Deduce that the line joining the midpoints of $AD$ and $BC$ also passes through $P$.